Some New Results Of (m, 4)-Splitting System And (m, 1, 2)-Separating System

Suppose m and t are integers such that 0 < t≤m. An (m, t)-Splitting System is a pair (X, B) that satisfies for every Y(?) X with |Y|=t, there is a subset B of X in B, such that | B∩Y|= (?) or | (X\ B)∩Y |=(?).Suppose m, t₁ and t₂ are integers such that t₁ + t₂≤m. An ( m,t₁,t₂ )-SeparatingSystem is a pair (X, B) which satisfies for every P(?) X, Q(?)X with |P|=t₁,|Q|=t₂ and P∩Q=Φ, there is a block B∈B for which either P(?)B,Q∩B=Φor Q(?)B,P∩B=Φ.In this paper, we first discuss some basic properties of the Splitting System and Separating System. We use powerful tools, the basic probabilistic method and greedy method to produce a bound of blocks size for the existence of both Splitting System and Separating System. We mainly concentrate on the case of both (m, 4)-Splitting System and (m, 1, 2)-SeparatingSystem.